Can the auto insurance company pay me directly for my medical expenses or do they have to pay the doctor?

I was in an auto accident and I have medical insurance. My medical insurance covered my medical expenses; however, I wanted to know if the auto insurance is still liable for those expenses? It seems as if my having medical insurance should benefit me and not the liable party's auto insurance company. If I did not have medical insurance the auto insurance company would still have to pay for those expenses. So shouldn't they still have to pay for those expenses and I get what ever is left that my medical insurance company did not pay? Are there any laws statures that back this up in Georgia? I am trying to settle this claim without getting an attorney and I need all the help I can get. Any advice would be helpful. I am in the process of writing a demand letter. Thanks for all the help.